Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,16)(14,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,45,47,44);(1,3,7,16)(2,8,12,13)(4,14,11,10)(5,6,15,9)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(4,11)(5,15)(6,9)(10,14)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,36,38,42)(34,39,41,35)(37,45,46,44)(40,43,48,47);(1,4,7,11)(2,9,12,6)(3,14,16,10)(5,13,15,8)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36)(34,35)(37,45)(38,42)(39,41)(40,47)(43,48)(44,46);(1,5,7,15)(2,10,12,14)(3,9,16,6)(4,8,11,13)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,37,38,46)(34,40,41,48)(35,43,39,47)(36,44,42,45);(1,6,7,9)(2,11,12,4)(3,15,16,5)(8,10,13,14)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957594);